#138351

order	512
nilpotency class	4
isoclinism	isoclinic class of #138198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10,12,17,25,31,37,35,14,16,24,30,13,18,26,33)(11,19,20,34,27,40,39,28,21,36,38,29,22,32,23,15);(1,3,4,7)(2,5)(6,8)(9,11,14,21)(10,15,16,28)(12,23,24,39)(13,27,25,22)(17,32,30,40)(18,34,31,29)(19,35,36,33)(20,37,38,26);(3,7)(5,8)(10,16)(12,24)(15,28)(18,31)(20,38)(23,39)(26,37)(29,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45547);